What is a poset called, when all elements with a lower bound, also share a upper bound?

Question

I have a partially ordered $M$ set that satisfies:

• For all $a, b, l \in M$, if $l \leq a$ and $l \leq b$ then there exists $u$, s.t. $a \leq u$ and $b \leq u$.

The property is strictly weaker than a lattice, even weaker than a union of lattices. Does it have a name?

What if the order is a pre-order?

Answer 1

This property is most commonly known as the confluence of $\leq$. It is used mostly in the context of term rewriting systems, but is occasionally used in other contexts as well; see, for instance, this math.SE question or this MathWorld page, found by Googling the term. Note that the stars which occasionally appear in these definitions are the reflexive-transitive closures of the relations being discussed, but the reflexive-transitive closure of an order relation is of course just that relation itself, as it is already reflexive and transitive.